Dual involutions in finite Coxeter groups

TL;DR

This paper explores the structure of involutions in finite Coxeter groups, focusing on their conjugacy classes and how products with the longest element behave, providing a classification framework.

Contribution

It identifies the conjugacy class of involution products with the longest element, extending the classification of involutions in finite Coxeter groups.

Findings

Classification of conjugacy classes of involutions

Identification of involution products with the longest element

Extension of known involution classifications

Abstract

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element is again an involution. We identify the conjugacy class of this product involution in terms of said classification.